Ten tens make a hundred

What is the middle zero doing in 105?

Ten ones make one ten, and ten tens make one hundred. Bundling does not change the total.

Earlier we bundled ten objects. Two bundles and three loose objects make twenty-three, written 23. We count tens and remaining ones rather than recounting every object.

Collect ten bundles of ten: there are one hundred objects. Exchange them for one hundred-bundle. No objects were created; only the size of a bundle changed.

Two hundreds, three tens and four ones are written 234. The 2 names two hundreds, the 3 three tens, and the 4 four ones. A digit’s place determines the amount it represents.

One hundred and five ones make 105. The middle zero records that there are no tens. Omitting it gives 15, a different amount. Zero holds a place even when that place contains no bundles.

Compare 105 and 150. Each has one hundred, but the rest is five ones versus five tens. Compare the largest places first, rather than just the last digit: 150 is greater.

Listen one paragraph at a time

1 / 5

2 / 5

3 / 5

4 / 5

5 / 5

If one bundle looked like one object

Draw a hundred-bundle unpacked into ten tens. More pictures now represent the same amount.

Now change it yourself

Change the bundles and exchange ten tens for a hundred. Check that the total is unchanged.

Choose a number

Hundreds 1

Tens 0

Ones 5

1×100 + 0×10 + 5 = 105

The grouping may change while the total stays fixed. Zero holds a place with no bundles.

This lesson uses whole numbers from 0 to 999. A bundle picture represents the number written on it, not one object.

Try with paper and objects

Draw bundle cards labelled 100, 10 and 1. You can reason with equal-value bundle drawings without collecting one hundred real objects.

Check your idea with another example

Check your idea 1

Three hundreds, no tens and seven ones. How do you write it?

Check your idea 2

Exchange ten tens for one hundred. What happens to the total?

Try a different setting

One shelf has 208 books and another 280. Which has more?

Write an answer yourself

Return to the explanation

Compare with an explanation

406 · There are no tens, so keep the middle zero: 406.

Does your explanation include these ideas?

Speak, point or draw on paper. Explain the reason, not just the numerical answer.

I named the value of each place.I explained why an exchange preserves the total.I gave an example where omitting zero changes the number.

Continue, or pause here

Add like places and explain why ten of a unit become one of the next unit.